These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.

Unless otherwise specified, all angles in this article are assumed to be in radians, but angles ending in a degree symbol (°) are in degrees. Per Niven's theorem multiples of 30° are the only angles that are a rational multiple of one degree and also have a rational sin/cos, which may account for their popularity in examples.[1]

The primary trigonometric functions are the sine and cosine of an angle. These are sometimes abbreviated sin(θ) and cos(θ), respectively, where θ is the angle, but the parentheses around the angle are often omitted, e.g., sin θ and cos θ.

The sine of an angle is defined in the context of a right triangle, as the ratio of the length of the side that is opposite to the angle divided by the length of the longest side of the triangle (the hypotenuse ).

The cosine of an angle is also defined in the context of a right triangle, as the ratio of the length of the side the angle is in divided by the length of the longest side of the triangle (the hypotenuse ).

The inverse trigonometric functions are partial inverse functions for the trigonometric functions. For example, the inverse function for the sine, known as the inverse sine (sin−1) or arcsine (arcsin or asin), satisfies

and

This article uses the notation below for inverse trigonometric functions:

By shifting the function round by certain angles, it is often possible to find different trigonometric functions that express particular results more simply. Some examples of this are shown by shifting functions round by π/2, π and 2π radians. Because the periods of these functions are either π or 2π, there are cases where the new function is exactly the same as the old function without the shift.

These are also known as the addition and subtraction theorems or formulae. They were originally established by the 10th century Persian mathematician Abū al-Wafā' Būzjānī. One method of proving these identities is to apply Euler's formula. The use of the symbols and is described in the article plus-minus sign.

For the angle addition diagram for the sine and cosine, the line in bold with the 1 on it is of length 1. It is the hypotenuse of a right angle triangle with angle β which gives the sin β and cos β. The cos β line is the hypotenuse of a right angle triangle with angle α so it has sides sin α and cos α both multiplied by cos β. This is the same for the sin β line. The original line is also the hypotenuse of a right angle triangle with angle α+β, the opposite side is the sin(α+β) line up from the origin and the adjacent side is the cos(α+β) segment going horizontally from the top left.

Overall the diagram can be used to show the sine and cosine of sum identities

The sum and difference formulae for sine and cosine can be written in matrix form as:

This shows that these matrices form a representation of the rotation group in the plane (technically, the special orthogonal groupSO(2)), since the composition law is fulfilled: subsequent multiplications of a vector with these two matrices yields the same result as the rotation by the sum of the angles.

In these two identities an asymmetry appears that is not seen in the case of sums of finitely many terms: in each product, there are only finitely many sine factors and cofinitely many cosine factors.

If only finitely many of the terms θi are nonzero, then only finitely many of the terms on the right side will be nonzero because sine factors will vanish, and in each term, all but finitely many of the cosine factors will be unity.

where ek is the kth-degree elementary symmetric polynomial in the n variables xi = tan θi, i = 1, ..., n, and the number of terms in the denominator and the number of factors in the product in the numerator depend on the number of terms in the sum on the left. The case of only finitely many terms can be proved by mathematical induction on the number of such terms. The convergence of the series in the denominators can be shown by writing the secant identity in the form

and then observing that the left side converges if the right side converges, and similarly for the cosecant identity.

A formula for computing the trigonometric identities for the third-angle exists, but it requires finding the zeroes of the cubic equation, where x is the value of the sine function at some angle and d is the known value of the sine function at the triple angle. However, the discriminant of this equation is negative, so this equation has three real roots (of which only one is the solution within the correct third-circle) but none of these solutions is reducible to a real algebraic expression, as they use intermediate complex numbers under the cube roots, (which may be expressed in terms of real-only functions only if using hyperbolic functions).

For specific multiples, these follow from the angle addition formulas, while the general formula was given by 16th century French mathematician Vieta.

In each of these two equations, the first parenthesized term is a binomial coefficient, and the final trigonometric function equals one or minus one or zero so that half the entries in each of the sums are removed. tan nθ can be written in terms of tan θ using the recurrence relation:

cot nθ can be written in terms of cot θ using the recurrence relation:

In particular, the formula holds when x, y, and z are the three angles of any triangle.

(If any of x, y, z is a right angle, one should take both sides to be ∞. This is neither +∞ nor −∞; for present purposes it makes sense to add just one point at infinity to the real line, that is approached by tan(θ) as tan(θ) either increases through positive values or decreases through negative values. This is a one-point compactification of the real line.)

For some purposes it is important to know that any linear combination of sine waves of the same period or frequency but different phase shifts is also a sine wave with the same period or frequency, but a different phase shift. This is useful in sinusoiddata fitting, because the measured or observed data are linearly related to the a and b unknowns of the in-phase and quadrature components basis below, resulting in a simpler Jacobian, compared to that of c and φ. In the case of a non-zero linear combination of a sine and cosine wave[dead link][29] (which is just a sine wave with a phase shift of π/2), we have

The following is perhaps not as readily generalized to an identity containing variables (but see explanation below):

Degree measure ceases to be more felicitous than radian measure when we consider this identity with 21 in the denominators:

The factors 1, 2, 4, 5, 8, 10 may start to make the pattern clear: they are those integers less than 21/2 that are relatively prime to (or have no prime factors in common with) 21. The last several examples are corollaries of a basic fact about the irreducible cyclotomic polynomials: the cosines are the real parts of the zeroes of those polynomials; the sum of the zeroes is the Möbius function evaluated at (in the very last case above) 21; only half of the zeroes are present above. The two identities preceding this last one arise in the same fashion with 21 replaced by 10 and 15, respectively.

Euclid showed in Book XIII, Proposition 10 of his Elements that the area of the square on the side of a regular pentagon inscribed in a circle is equal to the sum of the areas of the squares on the sides of the regular hexagon and the regular decagon inscribed in the same circle. In the language of modern trigonometry, this says:

In calculus the relations stated below require angles to be measured in radians; the relations would become more complicated if angles were measured in another unit such as degrees. If the trigonometric functions are defined in terms of geometry, along with the definitions of arc length and area, their derivatives can be found by verifying two limits. The first is:

verified using the identity tan(x/2) = (1 − cos x)/sin x. Having established these two limits, one can use the limit definition of the derivative and the addition theorems to show that (sin x)′ = cos x and (cos x)′ = −sin x. If the sine and cosine functions are defined by their Taylor series, then the derivatives can be found by differentiating the power series term-by-term.

Let i = √−1 be the imaginary unit and let denote composition of differential operators. Then for every odd positive integer n,

(When k = 0, then the number of differential operators being composed is 0, so the corresponding term in the sum above is just (sin x)n.) This identity was discovered as a by-product of research in medical imaging.[44]

When this substitution of t for tan(x/2) is used in calculus, it follows that sin(x) is replaced by 2t/(1 + t2), cos(x) is replaced by (1 − t2)/(1 + t2) and the differential dx is replaced by (2 dt)/(1 + t2). Thereby one converts rational functions of sin(x) and cos(x) to rational functions of t in order to find their antiderivatives.